Laminar heat transfer in a porous channel simulated with a two-energy equation model

Laminar heat transfer in a porous channel is numerically simulated with a two-energy equation model for conduction and convection. Macroscopic equations for continuity, momentum and energy transport for the fluid and solid phases are presented. The numerical methodology employed is based on the control volume approach with a boundary-fitted non-orthogonal coordinate system. Fully developed forced convection in a porous channel bounded by parallel plates is considered. Solutions for Nusselt numbers along the channel are presented for laminar flows. Results simulate the effects Reynolds number Re, porosity, particle size and solid-to-fluid thermal conductivity ratio on Nusselt sumber, Nu, which is defined for both the solid and fluid phases. High Re, low porosities, low particle diameters and low thermal conductivity ratios promote thermal equilibrium between phases leading to higher values of Nu.     

Large eddy simulation of flow over square cylinder arrays in an octagonal configuration at subcritical Reynolds numbers

In this study, fluid flow over an array of eight, 0.029 m × 0.029 m, square cross‐section cylinders in an octagonal configuration is studied numerically. The mean force coefficients (drag and lift) and the vortex formation characteristics of the array are calculated numerically by utilizing a three‐dimensional large eddy simulation mathematical model for turbulence. The numerical simulation is performed with commercial software ANSYS Fluent 19R1. To investigate the parametric influences, three spacings between the cylinders (0.07, 0.14, and 0.2 m), two array attack angles (0° and 15°), and two Reynolds numbers (4060 and 45 800) are considered. The results comprise flow patterns and force coefficients' variations with Reynolds numbers. The lift force of the downstream cylinder reaches its maximum at α = 15°, and the drag force of the upstream cylinders finds its peak at α = 0°. It is observed through velocity and viscosity contour plots that vortex formation length near the cylinder increases at higher Reynolds number. Velocity vector plots are also presented to show fluid flow behavior near the cylinder. Furthermore, the predicted mean forces on the cylinders are slightly different for different Reynolds numbers, spacings, and angles of attack.     

Thermal transport phenomena over a slot-perforated flat surface in pulsating free stream

A theoretical study is performed to investigate unsteady, two-dimensional, incompressible thermal-fluid flow over both sides of a slot-perforated flat surface, which is placed in a pulsating free stream. The effects of the pulsating Strouhal number, the Reynolds number Re, and the ratio of the slot width, d, to the plate thickness, δ, on the heat transfer performance and the velocity and thermal fields are disclosed. It is found from the study that: (i) when the free stream is pulsated, the alternating change in the fluid flow disturbs the thermal boundary layer formed along the plate and induces mixing of the upper and lower streams of the plate downstream from the slot, resulting in an amplification of heat-transfer performance; (ii) heat-transfer performance at the rear plate is induced with an increase in d/δ and Re; and (iii) heat transfer performance is intensified with an increase in fSr.     

Heat transfer enhancement in rectangular channels with axial ribs or porous foam under through flow and impinging jet conditions

Heat transfer and pressure loss characteristics of a high aspect ratio duct are measured under both, jet impingement and channel flow conditions, respectively. For both cases, roughness elements in consideration are staggered and inline axial ribs. The spacing (P) to height (e) ratios studied are P/e = 2 and P/e = 4; the rib height (e) to channel height (H) ratio is 0.125. Also studied is an aluminum foam roughness with a porosity of 92% and a height to channel height ratio of 0.15. Reynolds numbers considered for the channel flow case (based on the hydraulic diameter) range from 10,000 to 40,000. Reynolds numbers for the jet impingement case (based on the hole diameter) range from 5,000 to 20,000. Tests are performed using the copper plate regional average method. Results show a 50–90% increase in heat transfer due to the use of axial ribs in both, impingement and channel flow cases. The porous foam shows a more significant increase in heat transfer coefficient for both channel flow and impingement cases.     

Numerical computation of macroscopic turbulence quantities in representative elementary volumes of the porous medium

In this study, fully developed macroscopic turbulence quantities—based on their definitions in some existing turbulence models for porous media as well as those based on definitions introduced in a recently developed model [F.E. Teruel, Rizwan-uddin, A new turbulence model for porous media flows. Part I: Constitutive equations and model closure, Int. J. Heat Mass Transfer (2009)]—are computed and analyzed for different Reynolds numbers as well as for different porosity levels. When computed based on the definition introduced in the new model, these numerically computed, pore-level turbulent quantities provide closure to the formulation. A large set of microscopic turbulent flow simulations of the REV of a porous medium, formed by staggered square cylinders, is carried out to achieve these tasks. For each Reynolds number selected, ten different porosities are simulated in the 5–95% range. The Reynolds number is varied from Re = 10³ to Re = 10⁵, covering four different cases of the turbulence flow regime. Numerical results obtained for the macroscopic turbulent kinetic energy based on the new definition show that the spatial dispersion of the mean flow is the main contributor to this quantity at low porosities. Additionally, it is found that for high porosities, the spatial average of the turbulent kinetic energy is the main contributor but the spatial dispersion of the mean flow cannot be neglected. The new definition of the macroscopic dissipation rate is found to asymptotically approach the volume average of this quantity at high Reynolds numbers. It is confirmed that microscopic numerical simulations are consistent with the macroscopic law that states that the macroscopic dissipation rate is determined by the pressure-drop through the REV.     

Drag forces and heat transfer coefficients for spherical,cuboidal and ellipsoidal particles in cross flow at sub-critical Reynolds numbers

This work is devoted to the numerical calculation of heat and fluid flow past spherical particles and non-spherical particles of various shapes. Although numerous works have investigated drag forces (c_d) for spherical and non-spherical particles, works about the Nusselt number (Nu) relations for non-spherical particles are rare. Motivated by this fact, as a first step we consider cuboid, spherical and ellipsoidal particles in steady-state regimes corresponding to Reynolds numbers (Re) from 10 up to 250. Due to the asymmetric flow existing when Re approaches the value of 250, all simulations are made using a three-dimensional domain. Good agreement was observed when our numerical results gained for the sphere were compared with published values for drag coefficients and Nusselt numbers. Based on the analysis of numerical results obtained for non-spherical particles we found out that in addition to the Reynolds number three geometry parameters influence particle-fluid interaction: the drag coefficient depends primarily on the normalized longitudinal length, while both the sphericity and the crosswise sphericity influence the Nusselt number. For that reason new correlations are developed for both the drag coefficient and the Nusselt number. The accuracy of the closures developed for c_d and Nu is discussed in a comparison with published models.     

Numerical computation of nonlinear oscillatory two‐immiscible magnetohydrodynamic flow in dual porous media system: FTCS and FEM study

The transient Hartmann magnetohydrodynamic flow of two immiscible fluids flowing through a horizontal channel containing two porous media with oscillating lateral wall mass flux is studied. A two‐dimensional spatial model is developed for two fluids, one of which is electrically conducting and the other is electrically insulating. Both the fluid regimes are driven by a common pressure gradient. A Darcy‐Forchheimer drag force model is used to simulate the porous media effects on the flow in both the fluid regimes. Special boundary conditions are imposed at the interface. The governing second‐order nonlinear partial differential dimensionless equations are obtained for each region using a set of transformations. The resulting transport equations are controlled by the Hartmann hydromagnetic parameter (Ha), viscosity ratio parameter (α), two Darcy numbers (Da ₁ and Da ₂), two Forchheimer numbers (Fs ₁ and Fs ₂), two Reynolds numbers (Re ₁ and Re ₂), frequency parameter ( εA) associated with the transpiration (lateral wall flux) velocity and a periodic frequency parameter ( ω*t*). Numerical forward time/central space finite‐difference solutions are obtained for a wide range of the governing parameters. Bench marking is performed with a Galerkin finite‐element method (MAGNETO‐FEM), and the results are found to be in excellent agreement. Applications of the model include magnetic cleanup operations in coastal/ocean seabed oil spills and electromagnetic purification of petroleum reservoir fluids.     

Momentum and heat transfer characteristics of a semi-circular cylinder immersed in power-law fluids in the steady flow regime

The continuity, momentum and energy equations describing the flow and heat transfer of power-law fluids over a semi-circular cylinder have been solved numerically in the two-dimensional steady flow regime. The influence of the Reynolds number (Re), Prandtl number (Pr) and power-law index (n) on the local and global flow and heat characteristics have been studied over wide ranges of conditions as follows: 0.01 ? Re ? 30, 1 ? Pr ? 100 and 0.2 ? n ? 1.8. The variation of drag coefficient and Nusselt number with the Reynolds number, Prandtl number and power-law index is shown over the aforementioned ranges of conditions. In addition, streamline and isotherm profiles along with the recirculation length and distribution of pressure coefficient and Nusselt number over the surface of the semi-circular cylinder are also presented to gain further insights into the nature of the underlying kinematics. The wake size (recirculation length) shows almost linear dependence on the Reynolds number (Re ? 1) for all values of power-law index studied herein. The drag values show the classical inverse variation with the Reynolds number, especially for shear-thinning fluids at low Reynolds numbers. The point of maximum pressure coefficient is found slightly displaced from the front stagnation point for highly shear-thinning fluids, whereas for shear-thickening and Newtonian fluids, it coincides with the front stagnation point. For fixed values of the Prandtl number and Reynolds number, the rate of heat transfer decreases with the gradual increase in power-law index; this effect is particularly striking at high Prandtl numbers due to the thinning of the thermal boundary layer. Conversely, as expected, shear-thinning behavior facilitates heat transfer and shear-thickening impedes it. The effect of power-law index on both momentum and heat-transfer characteristics is seen to be appreciable at low Reynolds numbers and it gradually diminishes with the increasing Reynolds number.     

Characteristics for flow and heat transfer around a circular cylinder near a moving wall in wide range of low Reynolds number

The present study numerically investigates two-dimensional laminar fluid flow and heat transfer past a circular cylinder near a moving wall. Numerical simulations to calculate the fluid flow and heat transfer past a circular cylinder are performed for different Reynolds numbers varying in the range of 60–200 and a fixed Prandtl numbers of 0.7 (air) in the range of 0.1 ? G/D ? 4, where G/D is the ratio of the gap between the cylinder and a moving wall, G and the cylinder diameter, D. The flow and thermal fields become the steady state below the critical gap ratios of 0.8, 0.4 and 0.2 for the Reynolds numbers of 60, 80 and 100, respectively. As the gap ratio decreases, the magnitude of lift coefficient for all Reynolds numbers increased significantly with diminishing G/D due to the ground effect. The cases of Reynolds numbers of 60, 80 and 100 revealed the sharp slope of drag coefficient in the range of the gap ratio where the flow transfers from the unsteady state to the steady state. As the Reynolds number decreases, the variation of Nusselt is much significant and increases considerably with decreasing G/D.     

Mixed convection heat transfer in horizontal,concentric annuli for transitional flow conditions

Experiments have been performed to determine mixed convection flow and heat transfer in a horizontal, concentric tube annulus for Reynolds numbers in the range 2200 < Re < 5000. Within this range, flow conditions are turbulent and laminar, respectively, in regions of the annulus above and below the heated inner tube. For Reynolds numbers less than a critical value Re₁ which depends on the Rayleigh number, diameter ratio and longitudinal position, flow along the sides of the annulus is laminar and helicoidal. For Re >Re₁, there is a breakdown in the helicoidal motion, with subsequent transition to turbulence in the top and side regions of the annular passage. The local Nusselt number at the top of the inner tube is less than, equal to, and greater than that at the bottom for Re < Re₁, Re = Re₁, and Re >Re₁, respectively. The circumferentially averaged Nusselt number is weakly dependent on longitudinal position and may be correlated in terms of the Rayleigh and Reynolds numbers and the tube diameter ratio.     

Experimental and numerical characterization of the transverse dispersion at the exit of a short ceramic foam inside a pipe

The paper theoretically and numerically describes and experimentally studies transverse dispersion of a passive tracer in highly porous ceramic foams of different pore sizes. The pore Reynolds numbers range from 10 to 300. Digital images of the dispersion patterns were recorded and an approximate transverse dispersion coefficient was determined. Numerical solutions of the steady fluid flow and scalar concentrations confirm that the transverse dispersion coefficient models, based on the assumption of dominance of mechanical dispersion and on the linear dependence of the transverse dispersion model on ud, are able to predict satisfactorily the dispersion of a tracer for the range of Reynolds numbers considered. An alternative derivation of this linear dependence based on the closure of the volume averaged scalar transport equation is also presented. The influence of the length of the porous media in the stream direction on transversal and longitudinal dispersion is consistent with findings for packed beds at much lower Peclet and Reynolds numbers.     

Cocurrent turbulent mixed convection heat and mass transfer in falling film of water inside a vertical heated tube

A detailed numerical study has been conducted in order to analyse the combined buoyancy effects of thermal and mass diffusion on the turbulent mixed convection tube flows. Numerical results for air-water system are presented under different conditions. A low Reynolds number k-ε turbulent model is used with combined heat and mass transfer analysis in a vertical heated tube. The local heat fluxes, Nusselt and Sherwood numbers are reported to obtain an understanding of the physical phenomena. Predicted results show that a better heat transfer results for a higher gas flow Reynolds number Re, a higher heat flux q_w or a lower inlet water flow Γ₀. Additionally, the results indicate that the convection of heat by the flowing water film becomes the main mechanism for heat removal from the wall.     

Velocity field and turbulence effects on heat transfer characteristics from surfaces with V-shaped ribs

The flow field of smooth surfaces and surfaces with V-shaped ribs (V-SR) was studied experimentally with a Laser-Doppler Anemometry (LDA) system. In addition, heat transfer characteristics were experimentally investigated. Heat transfer results from these surfaces under impingement of a circular jet array (5 × 3) using an infrared thermal imaging technique are presented.The velocity profiles were measured at Reynolds number of 10,000 and at H/d equal to 3 and 12. For each H/d position, profiles were collected from x/d = 0 to 6 axial locations. The heat transfer data were obtained at Reynolds numbers equal to 2000, 6000, and 10,000. Along the target plate, different boundary layer profiles were obtained for smooth and V-SR plates at H/d = 3 and 12. Positions of maximum radial and axial velocities and turbulence intensities have been determined for smooth and V-SR plates. For low jet-to-plate spacings, the production of turbulence kinetic energy is higher for the V-SR surfaces as compared to smooth surfaces. For H/d = 3, the radial velocities are higher for the V-SR surfaces as compared to smooth surfaces but for H/d = 12, the radial velocities are not nearly changed all x/d locations. The heat transfer results have also been compared with those of a smooth surface under the same flow conditions to determine the enhancement in the heat transfer coefficient from x/d = 0 to 3 locations. In these locations, the Nusselt numbers are higher for the V-SR surfaces as compared to smooth surfaces. The locations of the peaks and the minima are influenced by cross flow velocities which in turn depend on jet-to-plate spacing and V-SR arrangements. For all results, the Nusselt numbers at the stagnation points decrease with increase in H/d.     

Numerical simulation of turbulent fluid flow and heat transfer characteristics in heat exchangers fitted with porous media

Numerical simulations have been carried out to investigate the turbulent heat transfer enhancement in the pipe filled with porous media. Two-dimensional axisymmetric numerical simulations using the k–? turbulent model is used to calculate the fluid flow and heat transfer characteristics in a pipe filled with porous media. The parameters studied include the Reynolds number (Re = 5000–15,000), the Darcy number (Da = 10^?1–10^?6), and the porous radius ratio (e = 0.0–1.0). The numerical results show that the flow field can be adjusted and the thickness of boundary layer can be decreased by the inserted porous medium so that the heat transfer can be enhanced in the pipe. The local distributions of the Nusselt number along the flow direction increase with the increase of the Reynolds number and thickness of the porous layer, but increase with the decreasing Darcy number. For a porous radius ratio less than about 0.6, the effect of the Darcy number on the pressure drop is not that significant. The optimum porous radius ratio is around 0.8 for the range of the parameters investigated, which can be used to enhance heat transfer in heat exchangers.     

Heat transfer in the magnetohydrodynamic flow of a power-law fluid past a porous flat plate with suction or blowing

An analysis is made of the steady magnetohydrodynamic flow of a power-law fluid past an infinite porous flat plate subjected to suction or blowing. A uniform transverse magnetic field is applied normal to the plate. It is shown that for small magnetic field parameter M, the steady solutions for velocity distribution exist for a pseudoplastic (shear-thinning) fluid for which the power-law index n satisfies 1/2 < n ≤ 1 provided that there is suction at the plate. For blowing at the plate the steady solutions for velocity distribution exist only when n is of the form p/q, where p is an odd positive integer and q is an even positive integer provided 1/2 < n < 1. Velocity at a point is found to increase with increase in M. The solution of the energy equation governing temperature distribution in the flow of a pseudoplastic fluid past an infinite porous plate subjected to uniform suction reveals that the temperature at a given point increases with increase in M.     

Flow and heat transfer in a driven square cavity with double-sided oscillating lids in anti-phase

Flow and heat transfer inside a square cavity with double-sided oscillating lids have been studied numerically. The oscillating angular frequency of lid motion, ?, and Reynolds number, Re, are two important parameters in this study. In terms of primary vortices, simulations at Re and ? up to 1000 and 5, respectively, showed that the flow patterns can be categorized into four modes: (i) a pair of vertical vortices, (ii) a pair of swing vortices, (iii) diagonal-dominated vortices and (iv) two pairs of swing vortices. The flow patterns change at different frequencies for Reynolds numbers greater than 300. Nevertheless, the oscillating frequency did not offer significant effect to change flow pattern at very low Reynolds number such as at Re ? 10. Heat transfer, represented by average Nusselt number (Nu) along the lids is increased at higher Re whereas it is decreased as ? increases.     

Milliscale confined impinging slot jets: Laminar heat transfer characteristics for an isothermal flat plate

Heat transfer and overall visualized flow characteristics of confined, laminar milli-scale slot jets are investigated, as they impinge upon an isothermal flat target plate, with a fully-developed profile at the nozzle exit. The effects of Reynolds number Re and normalized nozzle-to-plate distance ratio H/B are investigated for Re = 120–200, H/B = 2–10, and B = 1.0 mm, with a nozzle aspect ratio of y/B = 50. Instantaneous visualizations of overall slot jet flow structure show unsteady lateral distortions of jet columns at experimental conditions corresponding to the presence of continuous sinusoidal oscillations. Also apparent in flow visualization sequences are smoke signatures associated with instantaneous vortex structures which form as secondary flows develop in fluid which, initially, is just adjacent to the jet column. For each Reynolds number considered, local stagnation region Nusselt numbers Nu_o decrease dramatically as H/B increases to become greater than 7.2–13.2, as the Reynolds number is maintained constant at a value from 200 to 120, changes which occur just as continuous sinusoidal oscillation of the jet column begins to develop. The further development of continuous sinusoidal oscillating motion results in an approximate collapse of stagnation region Nusselt numbers measured at different Re and H/B values. When surface thermal boundary condition data are compared, the constant surface temperature data are generally higher than the constant surface heat flux data near the stagnation location, and lower at locations where x/B is greater than 1–2. The constant surface temperature data also show relatively low values with only very small changes with x/B, for x/B values which are greater than about 5.0. As such, the results illustrate the sensitivity of Nusselt numbers for laminar boundary layer and laminar slot jet flows to thermal boundary condition, as well as the restrictions on near-wall temperature gradients which result from a constant surface temperature thermal boundary condition.     

Two-dimensional mixed convection heat transfer from confined tandem square cylinders in cross-flow at low Reynolds numbers

A two-dimensional numerical simulation is carried out to understand the effects of thermal buoyancy and Prandtl number on flow characteristics and mixed convection heat transfer over two equal isothermal square cylinders placed in a tandem arrangement within a channel at low Reynolds numbers. The spacing between the cylinders is fixed with four widths of the cylinder. The numerical results are presented for the range of conditions as: 1 ≤ Re ≤ 30, 0.7 ≤ Pr ≤ 100 (the maximum value of Peclet number being 3000) and 0 ≤ Ri ≤ 1 for a fixed blockage parameter B = 10%. The unsteady numerical simulations are performed with a finite volume code based on the PISO algorithm in a collocated grid system. The representative streamlines, vortex structures and isotherm patterns are presented and discussed. In addition, the overall drag and lift coefficients, recirculation length and average Nusselt numbers are determined to elucidate the role of Reynolds, Prandtl and Richardson numbers on flow and heat transfer. It is found that the flow is completely steady for the chosen ranges of the parameters.     

Modelling of Non-Darcy Flows through Porous Media Using Extended Incompressible Smoothed Particle Hydrodynamics

In this article, a novel numerical method is presented for the simulation of non-Darcy flows through porous media by the incompressible smooth particle hydrodynamics (ISPH) method with a predictor-corrector scheme. In the ISPH algorithm, a semi-implicit velocity-correction procedure is used and the pressure is obtained by solving the pressure Poisson equation. The key point for the application to non-Darcy flows is to include porosity and drag forces of the medium (the Darcy term and the Forcheimer term) in the ISPH method. Unsteady lid-driven flow, natural convection in non-Darcy porous cavities, and natural convection at a porous medium–fluid interface are examined separately by our extended ISPH method. The results are presented with flow configurations, isotherms, and average Nusselt numbers for different Darcy numbers from 10^?4 to 10^?2, porosity values from 0.4 to 0.9, and Reynolds/Rayleigh numbers. The flow pattern and rate of heat transfer inside the cavity are affected by these parameters. The results demonstrate the important effect of the Darcy number on both the heat transfer rate and the flow regime. The results from this investigation are well validated and compare favorably with previously published results.     

Unsteady RANS simulation of turbulent flow and heat transfer in ribbed coolant passages of different aspect ratios

The flow and heat transfer in ribbed coolant passages of aspect ratios (AR) 1:1, 4:1, and 1:4 are numerically studied through the solution of the unsteady Reynolds averaged Navier–Stokes (URANS) equations. The URANS procedure, which utilizes a two equation k–ε model for the turbulent stresses, is shown to resolve large-scale bulk unsteadiness. The computations are carried out for a fixed Reynolds number of 25,000 and density ratio of 0.13, while the Rotation number is varied between 0.12 and 0.50.At higher rotation numbers (⩾0.5) at least three inter-rib modules are required to ensure periodicity in the streamwise direction. The flow exhibits unsteadiness in the Coriolis-driven secondary flow and in the separated shear layer. The average duct heat transfer is the highest for the 4:1 AR case. For this case, the secondary flow structures consist of multiple roll cells that direct flow both to the trailing and leading surfaces. The 1:4 AR duct shows flow reversal along the leading surface at high rotation numbers. For this AR, the potential for conduction-limited heat transfer along the leading surface is identified. The friction factor reveals an increase with the rotation number, and shows a significant increase at higher rotation numbers (∼Ro = 0.5).